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Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection

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  • Jianzhe Zhen

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg 5000 LE, Netherlands)

  • Dick den Hertog

    (Department of Econometrics and Operations Research, Tilburg University, Tilburg 5000 LE, Netherlands)

Abstract

We introduce a novel scheme based on a blending of Fourier-Motzkin elimination (FME) and adjustable robust optimization techniques to compute the maximum volume inscribed ellipsoid (MVE) in a polytopic projection. It is well-known that deriving an explicit description of a projected polytope is NP-hard. Our approach does not require an explicit description of the projection, and can easily be generalized to find a maximally sized convex body of a polytopic projection. Our obtained MVE is an inner approximation of the projected polytope, and its center is a centralized relative interior point of the projection. Since FME may produce many redundant constraints, we apply an LP-based procedure to keep the description of the projected polytopes at its minimal size. Furthermore, we propose an upper bounding scheme to evaluate the quality of the inner approximations. We test our approach on a simple polytope and a color tube design problem, and observe that as more auxiliary variables are eliminated, our inner approximations and upper bounds converge to optimal solutions.

Suggested Citation

  • Jianzhe Zhen & Dick den Hertog, 2018. "Computing the Maximum Volume Inscribed Ellipsoid of a Polytopic Projection," INFORMS Journal on Computing, INFORMS, vol. 30(1), pages 31-42, February.
    • Handle: RePEc:inm:orijoc:v:30:y:2018:i:1:p:31-42
    DOI: 10.1287/ijoc.2017.0763
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    References listed on IDEAS

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    1. den Hertog, Dick & Stehouwer, Peter, 2002. "Optimizing color picture tubes by high-cost nonlinear programming," European Journal of Operational Research, Elsevier, vol. 140(2), pages 197-211, July.
    2. Gorissen, Bram L. & Yanıkoğlu, İhsan & den Hertog, Dick, 2015. "A practical guide to robust optimization," Omega, Elsevier, vol. 53(C), pages 124-137.
    3. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    4. Zhen, Jianzhe & den Hertog, Dick, 2015. "Robust Solutions for Systems of Uncertain Linear Equations," Discussion Paper 2015-044, Tilburg University, Center for Economic Research.
    5. Zhen, Jianzhe & den Hertog, Dick, 2015. "Robust Solutions for Systems of Uncertain Linear Equations," Other publications TiSEM d072bdb9-4168-4522-90d8-1, Tilburg University, School of Economics and Management.
    6. Hendrix, Eligius M. T. & Mecking, Carmen J. & Hendriks, Theo H. B., 1996. "Finding robust solutions for product design problems," European Journal of Operational Research, Elsevier, vol. 92(1), pages 28-36, July.
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    3. ten Eikelder, Stefan, 2021. "Biologically-based radiation therapy planning and adjustable robust optimization," Other publications TiSEM 2654a17b-0a3c-4006-b644-e, Tilburg University, School of Economics and Management.

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